caa a22 4500 47573839X CHVBK 20180406123453.0 cr unu---uuuuu 170329e20000501xx s 000 0 eng 10.1007/s002220000059 doi (NATIONALLICENCE)springer-10.1007/s002220000059 The Bernstein problem for affine maximal hypersurfaces [Elektronische Daten] [Neil S. Trudinger, Xu-Jia Wang] Abstract. : In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R 2, must be a paraboloid. More generally, we shall consider the n-dimensional case, R n , showing that the corresponding result holds in higher dimensions provided that a uniform, "strict convexity” condition holds. We also extend the notion of "affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension n≥10. Springer-Verlag Berlin Heidelberg, 2000 Mathematics Subject Classification (1991): 53A15, 53A10, 53C45, 35J60 nationallicence Trudinger Neil S. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia, AU aut Wang Xu-Jia Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia, AU aut https://doi.org/10.1007/s002220000059 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s002220000059 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Trudinger Neil S. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia, AU aut NATIONALLICENCE 700 1- Wang Xu-Jia Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia, AU aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer